On embedded minimal disks in convex bodies
This paper is devoted to proving that for a bounded open strictly convex subset \(A\subset {\mathbb{R}}^ 3\) with \(C^ 4\) boundary, \(\partial A\), there exists an embedded minimal disc \(M\subset Clos(A)\) whose boundary \(\partial M\) is contained in \(\partial A\) and which is stationary (for the area integral) with respect to all variations preserving the inclusion \(\partial M\subset \partial A\), and the appropriate generalization to Riemannian manifolds is noted. The proof requires the use of methods developed in a number of different places, including the authors' own extension of W. Allard's regularity results to solutions of free boundary value problems. The argument begins by using the minimaxing methods of \textit{J. T. Pitts} [Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes 27 (1981; Zbl 0462.58003)] to obtain a minimaxing varifold which has a certain almost minimizing property. Then the methods for minimizing among embedded surfaces of \textit{F. Almgren} and \textit{L. Simon} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 447-495 (1979; Zbl 0417.49051)] and of \textit{W. Meeks}, \textit{L. Simon}, and \textit{S. T. Yau} [Ann. Math., II. Ser. 116, 621-659 (1982; Zbl 0521.53007)] are applied, together with curvature estimates extending those of \textit{R. Schoen} and \textit{L. Simon} [Commun. Pure Appl. Math. 34, 741-797 (1981; Zbl 0497.49034)] and the regularity result mentioned above, to show the almost minimizing varifold is actually an embedded minimal surface. Finally, using an argument of L. Simon and F. Smith, that embedded minimal surface is shown to be simply connected.
scientific article; zbMATH DE number 4000843